Essentials of Computational Chemistry

7.4 PERTURBATION THEORY 217
where A (0) is an operator for which we can find eigenfunctions, V is a perturbing operator,
and λ is a dimensionless parameter that, as it varies from 0 to 1, maps A (0) into A. Ifwe
expand our ground-state eigenfunctions and eigenvalues as Taylor series in λ, wehave
0 = (0)
0
0 + λ∂(0) +
∂λ
1
2! λ2 ∂2 (0)
0
∂λ2
+ 1
3! λ3 ∂3 (0)
0
∂λ3
+··· (7.20)
and
a0 = a (0)
0
λ=0
0
+ λ∂a(0)
∂λ
λ=0
λ=0
+ 1
2! λ2 ∂2a (0)
0
∂λ2
λ=0
λ=0
+ 1
3! λ3 ∂3a (0)
0
∂λ3
λ=0
+··· (7.21)
where a (0)
0 is the eigenvalue for (0)
0 , which is the appropriate normalized ground-state
eigenfunction for A (0) . For ease of notation, Eqs. (7.20) and (7.21) are usually written as
and
0 = (0)
0
a0 = a (0)
0
+ λ(1)
0 + λ2 (2)
0 + λ3 (3)
0 +··· (7.22)
+ λa(1)
0 + λ2a (2)
0 + λ3a (3)
0 +··· (7.23)
where the terms having superscripts (n) are referred to as ‘nth-order corrections’ to the zeroth
order term and are defined by comparison to Eqs. (7.20) and (7.21).
Thus, we may write
(A (0) + λV)|0〉 =a|0〉 (7.24)
as
(A (0) + λV)| (0)
0
(a (0)
0
+ λ(1)
0 + λ2 (2)
0 + λ3 (3)
0 +···〉=
+ λa(1)
0 + λ2a (2)
0 + λ3a (3)
0 +···)|(0) 0 + λ(1)
0 + λ2 (2)
0 + λ3 (3)
0 +···〉
(7.25)
Since Eq. (7.25) is valid for any choice of λ between 0 and 1, we can expand the left and
right sides and consider only equalities involving like powers of λ. Powers 0 through 3
require
A (0) | (0)
0 〉=a(0)
0 |(0)
0
A (0) | (1)
0 〉+V|(0)
0 〉=a(0)
0 |(1)
0 〉+a(1)
〉 (7.26)
0 |(0)
0
A (0) | (2)
0 〉+V|(1)
0 〉=a(0)
0 |(2)
0 〉+a(1)
0 |(1)
0 〉+a(2)
〉 (7.27)
0 |(0)
0
A (0) | (3)
0 〉+V|(2)
0 〉=a(0)
0 |(3)
0 〉+a(1)
0 |(2)
0 〉+a(2)
0 |(1)
0 〉+a(3)
〉 (7.28)
0 |(0)
0
〉 (7.29)
where further generalization should be obvious. Our goal, of course, is to determine the
various nth-order corrections. Equation (7.26) is the zeroth-order solution from which we
are hoping to build, while Eq. (7.27) involves the two unknown first-order corrections to the
wave function and eigenvalue.